Definition 3 (\(DD_n\)  and \(SDD_n\)  matrices [1]}) 
A symmetric matrix \(A=[a_{ij}]\)  is diagonally dominant (dd) if \(a_{ii} \ge \sum _{j \ne i} |a_{ij}|\) . A symmetric matrix A is scaled diagonally dominant (sdd) if there exists a diagonal matrix \(D\) , with positive diagonal entries, such that \(DAD\)  is dd. We denote the set of \(n \times n\)  dd (resp. sdd) matrices with \(DD_n \;(\text{resp.}~SDD_n)\) . Note that \( DD_n \subseteq SDD_n \subseteq \mathbb {S}^n_+\) .

